Melenberg, B.

English

Research Papers in Economics

Economics Letters 57 (1997) 1–4
Life cycle consumption models with uncertainty within periods
Bertrand Melenberg
Tilburg University, Dept. of Econometrics, P.O. Box 90153, 5000 LE Tilburg, The Netherlands
Received 13 March 1997; accepted 18 July 1997
Abstract
The standard life cycle model with a von Neumann-Morgenstern expected utility function is generalized to avoid
deterministic relationships in the multi-good version with intertemporal additive utility. The generalization can be estimated
and tested without complicating the econometric analysis. Ó 1997 Elsevier Science S.A.
Keywords: Life cycle model; Intertemporal additive utility
JEL classiﬁcation: C12
1. Introduction
When considering a life cycle model with intertemporal additive utility and with more than one
good per period, the ﬁrst order conditions not only result in Euler equations, but also in deterministic
relationships between marginal utilities pertaining to the same period. Such restrictions are clearly far
too strong, since they are not likely to be satisﬁed in any empirical application. Consequently, the life
cycle models must be speciﬁed such that these deterministic restrictions are relaxed.
The contribution of this paper is a modiﬁcation of the life cycle model that will accomplish this in a
natural way. The idea is to allow for much more variability across consumers in the way they plan
their consumption than usually is assumed. The modiﬁcation will be such that the resulting life cycle
model can be estimated and tested without complicating the econometric analysis. Moreover, the
proposed life cycle model is a generalization of the standard way of modelling life cycle models.
Consequently, it will also be possible to test whether the proposed extension makes sense.
The remainder of the paper is organized as follows. In Section 2 we discuss a standard formulation
of the life cycle model, together with the problem of the deterministic relationships and the standard
solution in terms of random preferences. In Section 3 we present our modiﬁcation and we discuss
some consequences of our modiﬁcation, in particular, the avoidance of deterministic relationships.
Finally, Section 4 contains some concluding remarks.
2. A standard life cycle model formulation
We start with a (ﬁnite) population S characterizing the consumers. In our model S will denote the
0165-1765/97/$17.00 Ó 1997 Elsevier Science S.A. All rights reserved.
PII S0165-1765(97)00184-52 B. Melenberg / Economics Letters 57 (1997) 1–4
population at the beginning of the ﬁrst period. We assume that there are L periods, with L [N the
1 maximum lifetime.We restrict attention to two goods per period . The ﬁrst one is always consumed in
positive quantities, the second one may be consumed only now and then. Consumer s [S is supposed
to solve in period 1 the following maximization problem with respect to q 5(q ,q )9,..., 11 1 2 1
q 5 ( q , q ) 9 : L1 L 2 L
L
Max E O u (s;q ) , subject to (1) HJ ( s ) tt
t 5 1
LL tt
2 1 2 1 OP (11r(s)) p (s)9q # a (s) 1OP (11r(s)) y (s), and SD SD j tt1 j t
j 5 1 j 5 1 t 5 1 t 5 1
q $ 0 t 5 1 ,..., L .( 2 ) 2 t
Here p (s) is the corresponding two-dimensional price vector of consumer s, u (s;.) is the utility index t t
of consumer s, y (s) denotes nominal non-property income in period t of consumer s, p (s) is the t t
nominal interest rate in period t of consumer s, a (s) is non-human wealth at the beginning of period 1
1 of consumer s, and E denotes taking expectation by consumer s at the beginning of period 1, (s)
conditional upon all information at that moment. In this model the vectors q are allowed to depend t
upon the variables contained in period t’s information set. Denote by v (s) the vector of random t
variables in period t’s information set of consumer s upon which q is allowed to depend. Included in t
v (s) are, for example, y (s) and p (s), for t#t. Over time, new information will be received, but old t tt
information will not be forgotten. Thus the vector v (s) contains the v (s) of all previous periods t,t. t t
Denote the optimal solution of optimization problem (1)-(2), as solved by consumer s,b y
( q ( s ) ,...,q( s ))9, with q (s) 5 (q (s),q (s))9. Then it is not hard to obtain the following standard 1 L t 1t 2t
Euler equations:
E h(­u /­q )/p (s)2(11r (s))(­u /­q )/p (s)j50, (3) (s,1) 1 11 11 2 2 12 12
E h[(­u ­q )/p (s)2(11r (s))(­u /­q )/p (s)]31( q ( s ))j 5 0, (4) (s,1) / 21 21 2 2 12 12 (0;`)2 1
with ­u /­q ;­u (s;q (s))/­q . Here E denotes the expectation of consumer s, conditional upon t itt t i t ( s ,1)
all information in period 1, i.e., conditional upon (s) and v (s). Restrictions (3)-(4) are not the only 1
ones. One can also derive as ﬁrst order condition
[(­u /­q )/p (s)2(­u /­q )/p (s)]31( q ( s )) 5 0. (5) 11 1 1 1 12 1 2 1 ( 0 ; ` )2 1
Contrary to (3)-(4), equation (5) is deterministic. It is, therefore, clearly far too strong: (5) will
generally not be satisﬁed in any empirical application! This indicates misspeciﬁcation of the model,
before even using (3)-(4).
Of course, result (5) is well known. A standard approach, applied in particular when the possibility
of binding nonnegativity constraints is ignored, is to introduce random preferences, see MaCurdy
(1983). However, as discussed by Adang and Melenberg (1995), a combination of random
preferences and nonnegativity constraints that may be binding is empirically unattractive. In order to
avoid (5), without loosing (4), some other approach is required.
1Generalization to more goods per period is straightforward.B. Melenberg / Economics Letters 57 (1997) 1–4 3
3. A life cycle model with ‘‘uncertainty within periods’’ and its consequences
˜˜ To modify the life cycle formulation, ﬁrst write v (s)5(v (s), v (s)), with v (s) the variables in the tt 2 1 tt
information set of period t, but not in the information set of period t21. If t51, we deﬁne
˜ ˜˜ v ( s ) 5 v ( s ). All we do is to allow for the possibility that q depends upon (v (s), v (s)), with v (s) 11 i tt 2 1 i t i t
˜˜ instead of v (s), where v (s) at least includes the price p (s), but not necessarily any other price or any t it it
other component of v (s). Thus, we simply allow for the possibility that different q 2s depend on t it
˜ different v (s)2s, which also may differ across consumers. We shall refer to this modiﬁcation by it
stating that we allow for ‘‘uncertainty within periods’’, or, perhaps even better ‘‘variation in
uncertainty within periods’’, where the variability applies to goods and consumers.
The motivation is simple. First, it is a straightforward and natural generalization of the standard
formulation. Intuition, without formalities, is provided by Adang and Melenberg (1995). Secondly,
without it, the standard life cycle model as presented in the previous section will be rejected as a
consequence of deterministic relationships. But with this modiﬁcation, the deterministic relationships
will not appear, as we will now show.
Notice ﬁrst that without variable uncertainty within periods (5) can be obtained using a calculus of
2 variation approach , with variations
h (s)51A ( s )3(1/p (s))1 (q (s)), (6) 11 11 (0;`)2 1
h ( s ) 52A ( s )3(1/p (s))1 (q (s)), (7) 21 21 (0;`)2 1
where h (s) corresponds to q (s) and h (s) corresponds to q (s), and where A(s) stands for the left 11 11 21 21
hand side of (5): add eh (s)t oq ( s ) and eh (s)t oq ( s ) and substitute these expressions in the 11 11 21 21
expected utility function for q (s) and q (s), respectively; then, by taking the derivative with respect 11 21
2 to e, and evaluating the resulting derivative at e50, we obtain: E (A(s)) 5 0, or A(s)50 (with (s)
3 probability one ). Of course, this is equation (5). In such a calculus of variations approach, only
directions are allowed which depend upon the same components of v (s) as the corresponding q (s). 1 i1
Without variable uncertainty within periods, both h (s) and h (s) are allowed to depend upon the 11 21
˜ whole of v (s). But with variable uncertainty within periods, h (s) is only allowed to depend on v (s) 1 11 11
˜ and h (s) is only allowed to depend on v (s). But then the directions h (s) and h (s) given in (6) 21 21 11 21
˜˜ and (7) are no longer valid directions: A(s) depends on both v (s) and v (s), and q (s) in (7) also 11 21 21
˜ depends on v (s). But if the directions (6) and (7) are not allowed any more, we are also no longer 21
able to obtain the deterministic relationship (5)!
Consider next (3). Choose as variations
h (s)5B(s)/p (s), h (s)52B ( s )(11r (s))/p (s), (8) 11 11 12 2 12
with B(s) the left hand side of (3), but not using E , but, instead, E : the expectation conditional (s,1) (s,11)
˜ upon (s) and v (s). Using these variations, we obtain (with probability one) 11
2See Hall (1978) for the use of calculus of variations in life cycle models and see Neustadt (1976) for an extensive study on
optimization and ﬁrst order conditions.
3Here and in the sequel ‘‘with probability one’’ means with probability one with respect to the probability distribution
corresponding to E . (s)4 B. Melenberg / Economics Letters 57 (1997) 1–4
E ((­u /­q )/p (s)2(11r (s))(­u /­q )/p (s)) 5 0, (9) (s,11) 1 11 11 2 2 12 12
Similarly, we obtain as alternative of (4) (with probability one)
E ([(­u /­q )/p (s)2(11r (s))(­u /­q )/p (s)]31( q ( s ))) 5 0, (10) (s,12) 1 21 21 2 2 12 12 (0;`)2 1
˜ where E stands for the expectation conditional on (s) and v (s). (s,12) 12
Thus, we see that the Euler equations (3)-(4) remain valid, with only a slight modiﬁcation: the
conditioning should not be on (s) and the whole of v (s), but on (s) and a part of v (s), depending upon 11
the Euler equation under consideration. The Euler equations (9)-(10) provide an easy rule of thumb.
The part of v (s) is generally only the price of the ﬁrst period’s good that occurs in the Euler equation: 1
˜ according to our assumptions at least p (s) is included in v (s), and since no other information 11 i1
˜ concerning the correct form of v (s) is available, the rule of thumb follows. i1
4. Concluding remarks
For empirical analyses unconditional moment restrictions can be obtained by using instruments that
are functions of s and the admissible part of v (s) which follows from the given rule of thumb. Thus, 1
due to within period variable uncertainty, not all variables included in v (s) can be used as 1
instruments. In particular, variables like r (s) and y (s) are excluded. This has an important 11
consequence: excess sensitivity with respect to these latter variables need not imply rejection of the
life cycle model presented in this paper. In other words, rejection of our version of the life cycle
model becomes much harder than when using the standard way of modelling, although the
econometric part is not complicated. This is already illustrated by Adang and Melenberg (1995).
Acknowledgements
Pim Adang, Rob Alessie, Arie Kapteyn, Marcel Kerkhofs and Arthur van Soest provided helpful
comments.
References
Adang, P., Melenberg, B., 1995. Nonnegativity constraints and intratemporal uncertainty in a multi-good life cycle model.
Journal of Applied Econometrics, 10, 1–15.
Hall, R.E., 1978. Stochastic implications of the life cycle permanent income hypothesis: Theory and evidence. Journal of
Political Economy, 86, 971–988.
MaCurdy, T., 1983. A simple scheme for estimating an intertemporal model of labor supply and consumption in the presence
of taxes and uncertainty. International Economic Review, 24, 265–289.
Neustadt, L.W., 1976. Optimization: A Theory of Necessary Conditions. Princeton University Press, NJ.

Life cycle consumption models with uncertainty within periods.

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